1070
IEEE COMMUNICATIONS LETTERS, VOL. 25, NO. 4, APRIL 2021
Parameters of Hulls of Primitive BCH Codes of Length q 3 − 1
Chunyu Gan, Chengju Li , and Haifeng Qian
Abstract— Let C(q,n,δ) be the primitive BCH code over Fq of
length n = q 3 −1 with designed distance δ, where Fq is the finite
field of order q. In this letter, we investigate the parameters of the
codes C(q,n,δ) and their hulls for 2 ≤ δ ≤ n. We also present a
sufficient and necessary condition on the designed distance such
that the hull has the largest dimension.
hulls for 2 ≤ δ ≤ q 3 − 1. We also present a sufficient and
necessary condition on the designed distance such that the hull
has the largest dimension. Since it is more complex to deal
with the cyclotomic cosets in non-primitive case, we focus on
the primitive BCH codes here.
Index Terms— Linear code, BCH code, self-orthogonal code,
hull, cyclotomic coset.
II. P RELIMINARIES
I. I NTRODUCTION
ET Fq be the finite field of order q, where q is a power
of a prime p. An [n, k, d] linear code C over Fq is
a k-dimensional subspace of Fnq with minimum (Hamming)
distance d. The (Euclidean) dual code of C is defined by
L
C ⊥E = {b ∈ Fnq |
n−1
bi ci = 0 ∀ c ∈ C},
i=0
where b = (b0 , b1 , . . . , bn−1 ) and c = (c0 , c1 , . . . , cn−1 ). The
hull of the linear code C is defined to be
HullE (C) := C ∩ C ⊥E .
It is clear that the hull is a self-orthogonal linear code over Fq .
The concept of the hull was originally introduced in 1990 by
Assmus and Key [1] to classify finite projective planes. It was
shown that the hull plays an important role in constructing
entanglement assisted quantum error correcting codes [11],
[16]. It is then important and interesting to investigate the
hulls of linear codes and obtain self-orthogonal codes.
BCH codes are a special class of linear codes and have
nice structures. Recently, several sufficient and necessary
conditions for BCH codes to have large hulls were developed
and self-orthogonal codes were presented [10]. In this letter,
we determine the dimensions of the codes C(q,q3 −1,δ) and their
Manuscript received October 28, 2020; revised November 27, 2020;
accepted December 5, 2020. Date of publication December 9, 2020; date
of current version April 9, 2021. The work of Chengju Li was supported by
the National Natural Science Foundation of China (12071138, 11701179),
the Shanghai Chenguang Program (18CG22), and the Key Laboratory of
Applied Mathematics of Fujian Province University (SX201903). The work of
Haifeng Qian was supported by National Natural Science Foundation of China
(61961146004, 61632012). The associate editor coordinating the review of this
letter and approving it for publication was M. Battaglioni. (Corresponding
author: Chengju Li.)
Chunyu Gan and Chengju Li are with the Shanghai Key Laboratory of
Trustworthy Computing, East China Normal University, Shanghai 200062,
China, and also with the Key Laboratory of Applied Mathematics (Putian
University), Fujian Province University, Putian 351100, China (e-mail:
51194501119@stu.ecnu.edu.cn; cjli@sei.ecnu.edu.cn).
Haifeng Qian is with the Software Engineering Institute, East China Normal
University, Shanghai 200062, China, and also with the Shanghai Institute
of Intelligent Science and Technology, Tongji University, Shanghai 200092,
China (e-mail: hfqian@admin.ecnu.edu.cn).
Digital Object Identifier 10.1109/LCOMM.2020.3043448
Let Zn = {0, 1, 2, . . . , n−1} be the ring of integers modulo
n. For any s ∈ Zn , the q-cyclotomic coset of s modulo n is
defined by
Cs = {s, sq, sq 2 , . . . , sq ls −1 } mod n ⊆ Zn ,
where ls is the smallest positive integer such that s ≡ sq ls
(mod n), and is the size of the q-cyclotomic coset. The
smallest integer in Cs is called the coset leader of Cs , and
we use Γ(n,q) to denote the set of all coset leaders modulo n.
A linear code C is said to be cyclic if (c0 , c1 , . . . , cn−1 ) ∈ C
implies (cn−1 , c0 , . . . , cn−2 ) ∈ C. It is well-known that C =
g(x) , where g(x) is a monic polynomial over Fq with the
smallest degree and g(x) | (xn − 1) [18]. Then g(x) is called
the generator polynomial and h(x) = (xn −1)/g(x) is referred
to as the check polynomial of C.
For n = q m − 1, let α be a generator of F∗qm . The set
T = {0 ≤ i ≤ n − 1 | g(αi ) = 0} is referred to as the
defining set of C. In fact, T is a union of some q-cyclotomic
cosets modulo n [18]. For 0 ≤ i ≤ n− 1, let mi (x) denote the
minimal polynomial of αi over Fq . For an integer δ ≥ 2, define
g(q,n,δ) (x) = lcm m1 (x), m2 (x), . . . , mδ−1 (x) ,
where lcm denotes the least common multiple of these polynomials. Let C(q,n,δ) be the cyclic code of length n with
generator polynomial g(q,n,δ) (x). Then C(q,n,δ) is called a
primitive narrow-sense BCH code with designed distance δ.
This means that the minimum distance of C(q,n,δ) is greater
than or equal to the designed distance δ. Charpin pointed
out in [5] that it is a well-known hard problem to determine
the minimum distance of BCH codes. We refer the reader to
[2]–[4], [6]–[9], [12]–[15], [17] and references therein for
known results on BCH codes.
III. H ULLS OF P RIMITIVE BCH C ODES OF L ENGTH q 3 − 1
In this section, we always assume that n = q 3 − 1 and
C(q,n,δ) is the primitive narrow-sense BCH code over Fq with
designed distance δ. We aim to investigate the parameters of
C(q,n,δ) and HullE (C(q,n,δ) ), and determine the maximal value
of dim(HullE (C(q,n,δ) )) when δ runs from 2 to n. Let Ci be
a q-cyclotomic coset modulo n. It is easy to get the following
lemma, which will be employed later.
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GAN et al.: PARAMETERS OF HULLS OF PRIMITIVE BCH CODES OF LENGTH Q3 − 1
Lemma 3.1: Let T be a union of some q-cyclotomic cosets
modulo n. Let i be a coset leader of the q-cyclotomic coset
Ci and Ci T . Denote Z = T ∪ Ci and T −1 =
{n − t mod n | t ∈ T }. Then the following statements hold.
1) If Ci = Ci−1 , then |Z ∪ (Zn \ Z −1 )| = |T ∪ (Zn \ T −1)|.
2) If Ci = Ci−1 and Ci−1 T , then |Z ∪ (Zn \ Z −1 )| =
|T ∪ (Zn \ T −1 )| − |Ci |.
3) If Ci = Ci−1 and Ci−1 ⊆ T , then |Z ∪ (Zn \ Z −1 )| =
|T ∪ (Zn \ T −1 )| + |Ci |.
Let i ≥ 2 be a coset leader and T = C1 ∪ C2 ∪ · · · ∪ Ci−1 ,
where Cj is the q-cyclotomic coset modulo n. Define μ1 = 1
and
⎧
⎪
if Ci = Ci−1 and Ci−1 T ;
⎨1,
μi = 0,
if Ci = Ci−1 ;
⎪
⎩
−1, if Ci = Ci−1 and Ci−1 ⊆ T.
Let kδ be the dimension of HullE (C(q,n,δ) ) and denote I =
i ∈ Γ(n,q) | 1 ≤ i ≤ δ − 1 . For δ = 2, the defining set of
C(q,n,δ) is C1 , while the defining set of HullE (C(q,n,δ) ) is C1 ∪
(Zn \ C1−1 ). It is easy to check that C1 = C1−1 and C1 ⊆
(Zn \ C1−1 ). Thus k2 = n − (n − |C1−1 |) = |C1 | = μ1 |C1 |.
We then obtain that
μi |Ci |
(1)
kδ = n − |T ∪ (Zn \ T −1 )| =
i∈I
from Lemma 3.1 and the definition of μi by induction. Thus
the values of μi are necessary to determine the dimension of
the hull. Let s be the coset leader of Ci−1 . It is then deduced
that μi = 1 if s > i, μi = −1 if s < i, and μi = 0 if s = i.
Furthermore, one can check that μi = 0 if q is even.
Proposition 3.2: Let i ≥ 2 be a coset leader. Write i =
i2 q 2 + i1 q + i0 = (i2 , i1 , i0 ) be the q-adic expansion. Then
the following statements hold.
1) When i2 ≤ i1 < q − 1 − i2 , we have
μi =
1,
−1,
if i2 ≤ i0 < q − i2 ;
if q − i2 ≤ i0 ≤ q − 1.
2) When q − 1 − i2 ≤ i1 ≤ q − 1, we have
μi =
0,
−1,
if q is odd and i2 = i1 = i0 =
otherwise.
q−1
2 ;
Proof: Note that i is a coset leader. One can check that
i2 ≤ i1 and i2 ≤ i0 . In fact, i2 < i0 if i2 < i1 and i2 ≤ i0
if i2 = i1 . It is clear that Ci−1 = {[−i]n , [−iq]n , [−iq 2 ]n },
where [j]n = j mod n. Write s as the coset leader of Ci−1 .
Note that
[−i]n = (q − 1 − i2 , q − 1 − i1 , q − 1 − i0 ),
[−iq]n = (q − 1 − i1 , q − 1 − i0 , q − 1 − i2 ),
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If q − i2 ≤ i0 ≤ q − 1, we have q − 1 − i0 ≤ i2 − 1 < i2
and [−iq 2 ]n < i. Thus μi = −1.
2) When q − 1 − i2 ≤ i1 ≤ q − 1, we have q − 1 − i1 ≤ i2 .
• If i2 < i0 , we have q − 1 − i0 < q − 1 − i2 ≤ i1
and s ≤ [−iq]n < i. Then μi = −1.
• If i2 = i0 , we have i2 = i1 = i0 by noticing that i
is a coset leader. Note that q − 1 − i2 ≤ i1 ≤ q − 1,
we then get i2 ≥ q−1
2 . When q is even, it is easy
to see that s = [−i]n < i and μi = −1. When
q is odd, we have s = [−i]n = i if and only if
i2 = i1 = i0 = q−1
2 , and μi = 0. In addition,
if i2 > q−1
,
it
is
easy
to check that s < i and
2
μi = −1.
Concluding all discussions above, we get the desired
conclusion.
Let δ2 − 1 = j2 q 2 + j1 q + j0 be a coset leader with δ2 > δ1 ,
where δ1 = j2 q 2 with 1 ≤ j2 < q2 . Let kδ1 and kδ2 be
the dimensions of HullE (C(q,q3 −1,δ1 ) ) and HullE (C(q,q3 −1,δ2 ) ),
respectively. The following proposition gives an important
relation between kδ1 and kδ2 .
Proposition 3.3: With the notation and assumptions above,
we have the following.
1) When j2 ≤ j1 < q − 1 − j2 and j2 ≤ j0 < q − j2 ,
we have
kδ2 = kδ1 + 9j22 − 3qj2 + (3q − 3 − 9j2 )j1 + 3j0 + 1.
2) When j2 ≤ j1 < q − 1 − j2 and q − j2 ≤ j0 ≤ q − 1,
we have
kδ2 = kδ1 +9j22 −(3q+6)j2 +(3q−3−9j2)j1
−3j0 +6q−5.
3) When q − 1 − j2 ≤ j1 ≤ q − 1, we have
kδ2 = kδ1 + 21j22 + (24 − 21q)j2
+(3j2 − 3q + 3)j1 − 3j0 + 6q 2 − 12q + 7.
Proof: Write i = (i2 , i1 , i0 ). One can easily see that
|Ci | = 1 if and only if i2 = i1 = i0 .
1) When j2 ≤ j1 < q − 1 − j2 and j2 ≤ j0 < q − j2 , denote
I = {i ∈ Γ(n,q) |δ1 ≤ i ≤ δ2 − 1}.
•
Denote I1 = {i ∈ I | i = (j2 , j2 , j2 )}. For i ∈ I1 ,
we have μi = 1 by Proposition 3.2 and |Ci | = 1.
It is clear that |I1 | = 1 and then
μi |Ci | = 1.
•
Denote I2 = {i ∈ I | (j2 , i1 , j2 + 1) ≤ i ≤
(j2 , i1 , q − j2 − 1) with j2 ≤ i1 ≤ j1 − 1}. For
i ∈ I2 , we have μi = 1 by Proposition 3.2 and
|Ci | = 3. Since |I2 | = (q−1−2j2)(j1 −j2 ), we have
μi |Ci | = 3(q − 1 − 2j2 )(j1 − j2 ).
i∈I1
i∈I2
•
[−iq 2 ]n = (q − 1 − i0 , q − 1 − i2 , q − 1 − i1 ).
1) When i2 ≤ i1 < q − 1 − i2 , we have q − 1 − i2 > i2
and q − 1 − i1 > i2 . If i2 ≤ i0 < q − i2 ,
we have q − 1 − i0 ≥ i2 . It is then deduced that
s = min{[−i]n , [−iq]n , [−iq 2 ]n } > i. Thus μi = 1.
Denote I3 = {i ∈ I | (j2 , i1 , q − j2 ) ≤ i ≤
(j2 , i1 , q − 1) with j2 ≤ i1 ≤ j1 − 1}. For i ∈ I3 ,
we have μi = −1 by Proposition 3.2 and |Ci | = 3.
μi |Ci | =
Since |I3 | = j2 (j1 − j2 ), we have
i∈I3
•
−3j2 (j1 − j2 ).
Denote I4 = {i ∈ I | (j2 , j1 , j2 + 1) ≤ i ≤
(j2 , j1 , j0 )}. For i ∈ I4 , we have μi = 1 by
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1072
IEEE COMMUNICATIONS LETTERS, VOL. 25, NO. 4, APRIL 2021
Proposition 3.2 and |Ci | = 3. It is clear that |I4 | =
j0 − j2 , we have
μi |Ci | = 3(j0 − j2 ).
i∈I4
One can check that I = I1 ∪ I2 ∪ I3 ∪ I4 is a disjoint
union. Then we have
kδ2 = kδ1 +
= kδ1 +
i∈I
9j22
μi |Ci | = kδ1 +
4
μi |Ci |
s=1 i∈Is
− 3qj2 + (3q − 3 − 9j2 )j1 + 3j0 + 1.
2) When j2 ≤ j1 < q − 1 − j2 and q − j2 ≤ j0 ≤ q − 1,
denote
I = {i ∈ Γ(n,q) | δ ≤ i ≤ δ2 − 1},
where δ − 1 = (j2 , j1 , q − j2 − 1). Then we have
kδ = kδ1 + 9j22 − (3q + 3)j2
+(3q − 3 − 9j2 )j1 + 3q − 2.
For i ∈ I , we have μi = −1 and |Ci | = 3. Moreover,
|I | = δ2 − δ = j2 + j0 − q + 1 and
μi |Ci | =
i∈I −3(j2 + j0 − q + 1). By Equation (1), we have
μi |Ci | = kδ1 + 9j22 − (3q + 6)j2
kδ2 = kδ +
i∈I +(3q − 3 − 9j2 )j1 −3j0 +6q−5.
3) When q − 1 − j2 ≤ j1 ≤ q − 1, the proof is very similar
to those of the former cases and we omit the details.
Concluding all discussions above, we get the desired
conclusion.
Below we will determine the dimension of
HullE (C(q,q3 −1,δ) ), where δ has the following two cases: (1)
2 ≤ δ ≤ q 2 ; (2) δ = rq 2 with 2 ≤ r < q2 .
Proposition 3.4: Let δ − 1 = (0, j1 , j0 ) be a coset leader,
where 2 ≤ δ ≤ q 2 . The dimension of HullE (C(q,q3 −1,δ) ) is
given as follows.
1) When j2 ≤ j1 < q − 1 and j2 ≤ j0 ≤ q − 1, we have
kδ = (3q − 3)j1 + 3j0 .
2) When j1 = q − 1, we have kδ = −3j0 + 3q 2 − 6q + 3.
Proof: The proof is very similar to that of Proposition 3.3
by letting j2 = 0 and deleting some cases of Is , so we omitted
the details here.
Proposition 3.5: When δ = rq 2 with 2 ≤ r < q2 , we have
kδ = 7r3 − 9qr2 + 3q 2 r − 1.
Proof: Let I(s) = {i ∈ Γ(n,q) | sq 2 ≤ i < (s + 1)q 2 },
where 1 ≤ s < q2 . By Proposition 3.3, we have
μi |Ci | = 21s2 + (21 − 18q)s + 3q 2 − 9q + 7.
i∈I(s)
In addition, it is easy to see that
μi |Ci | = kq2 = 3q 2 − 9q + 6,
i∈I(0)
where I(0) = {i ∈ Γ(n,q) | 1 ≤ i < q 2 }. Note that {i ∈
Γ(n,q) | 1 ≤ i < rq 2 } = I(0) ∪ I(1) ∪ · · · ∪ I(r − 1) is
a disjoint union. Then
kδ =
i∈I(0)
3
μi |Ci | +
r−1 μi |Ci |
s=1 i∈I(s)
2
= 7r − 9qr2 + 3q r − 1.
The proof is completed.
Let δ − 1 = (j2 , j1 , j0 ) be a coset leader, where
0 ≤ j2 < q2 . The following theorem gives the dimension of
the hull of the code C(q,q3 −1,δ) .
Theorem 3.6: Let n = q 3 − 1 and let δ − 1 = (j2 , j1 , j0 )
be a coset leader, where 0 ≤ j2 < q2 . Then the dimension
kδ is given as follows.
1) When j2 ≤ j1 < q − 1 − j2 and j2 ≤ j0 < q − j2 ,
we have
kδ = 7j23 + (9 − 9q)j22
+(3q 2 − 3q)j2 + (3q − 3 − 9j2 )j1 + 3j0 .
2) When j2 ≤ j1 < q − 1 − j2 and q − j2 ≤ j0 ≤ q − 1,
we have
kδ = 7j23 + (9 − 9q)j22 + (3q 2 −3q−6)j2
+(3q − 3 − 9j2 )j1 − 3j0 +6q−6.
3) When q − 1 − j2 ≤ j1 ≤ q − 1, we have
kδ = 7j23 + (21 − 9q)j22 + (3q 2 − 21q + 24)j2
+(3 − 3q + 3j2 )j1 − 3j0 + 6q 2 − 12q + 6.
Proof: It is straightforward to get the desired conclusion
from Propositions 3.3, 3.4, and 3.5.
Let δ − 1 = (j2 , j1 , j0 ) be a coset leader, where j2 ≥ q2 .
We can similarly get the following theorem on the dimension
of the hull of the code C(q,q3 −1,δ) and omit the proof here.
Theorem 3.7: Let n = q 3 − 1 and let δ − 1 = (j2 , j1 , j0 )
be a coset leader, where j2 ≥ q2 . Then we have
kδ = −j23 + (3q − 3)j22 + (3q − 3q 2 )j2
+(3 − 3q + 3j2 )j1 − 3j0 + q 3 − 2.
When q ≤ 5, the parameters of the hulls with largest
dimension are given as follows.
Next we determine the designed distance such that the hull
of a primitive narrow-sense BCH code with length n = q 3 − 1
over Fq has the largest dimension, where q ≥ 7. For a fixed
2
j2 in the range 1 ≤ j2 < q−1
3 and j2 q < δ ≤ (j2 +
2
1)q , it is easy to get from Theorem 3.6 that the dimension of
HullE (C(q,q3 −1,δ) ) is maximal if and only if δ has the form
δ = (j2 + 1)(q 2 − q − 1) + 1.
2
Furthermore, if δ > q−1
3 q , one can check from
Theorems 3.6 and 3.7 that kδ < kc(q2 −q−1)+1 , where
2
c = q−1
3 . If 2 ≤ δ ≤ q , then it can be similarly checked
that kδ < k2(q2 −q−1)+1 .
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GAN et al.: PARAMETERS OF HULLS OF PRIMITIVE BCH CODES OF LENGTH Q3 − 1
The following theorem gives a sufficient and necessary
condition on the designed distance that HullE (C(q,q3 −1,δ) ) has
the largest dimension for 2 ≤ δ ≤ n.
Theorem 3.8: Let q ≥ 7 be a prime power and 2 ≤ δ ≤ n.
Then the dimension of HullE (C(q,q3 −1,δ) ) is maximal if and
only if
δ = (t + 1)(q 2 − q − 1) + 1,
√
18q−15− 72q2 −36q−363
. Furthermore, the selfwhere t = 42
orthogonal code HullE (C(q,q3 −1,δ) ) has parameters [q 3 −
1, k, ≥ δ + 1], where k = 7t3 + (18 − 9q)t2 + (3q 2 − 15q +
18)t + 3q 2 − 6q + 3.
Proof: Denote kδi = dim(HullE (C(q,q3 −1,δi ) )), where
δi = (i + 1)(q 2 − q − 1) + 1 and 1 ≤ i < q−1
3 . Then
the largest dimension of HullE (C(q,q3 −1,δ) ) is equal to
max kδ1 , kδ2 , . . . , kδ q−1
.
3
−1
Denote βi = kδi − kδi−1 for 2 ≤ i < q−1
3 . Then βi =
21i2 + (15 − 18q)i + 3q 2 − 6q + 7 by Theorem 3.6. Moreover,
it is easy to check that βi = 0 since βi ≡ 1 (mod 3).
Thus we have
√ βi > 0 if and only if 2 ≤ i ≤ t, where
18q−15−
72q2 −36q−363
.
42
2
t=
When δ = (t + 1)(q − q − 1) + 1, the dimension
of HullE (C(q,q3 −1,δ) ) can be obtained from Theorem 3.6.
Furthermore, the set {0, 1, 2, . . . , δ − 1} is contained in the
defining set of HullE (C(q,q3 −1,δ) ) and the desired conclusion
on minimum distance then follows from the BCH bound.
Example 1: The parameters of some hulls with the largest
dimension are given as follows.
As a conclusion, we present the dimension of the BCH
code C(q,n,δ) .
Theorem 3.9: Let n = q 3 − 1 and let δ − 1 = (j2 , j1 , j0 )
be a coset leader. Then the BCH code C(q,n,δ) has parameters
[n, k, d ≥ δ], where
k = n − [j23 + (3 − 3q)j22 + (3q 2 − 3q)j2
+(3q − 3 − 3j2 )j1 + 3j0 ].
Proof: Denote i = (i2 , i1 , i0 ). It is easy to see that
|Ci | = 1 if and only if i2 = i1 = i0 . Denote I = {i ∈
Γ(n,q) | 1 ≤ i ≤ δ − 1}. Then we have the following four
cases.
1) Denote I1 = {i ∈ Γ(n,q) | i = (i2 , i2 , i2 ) with 1 ≤ i2 ≤
j2 }. For i ∈ I1 , we have |Ci | = 1 and |I1 | = j2 . Then
|Ci | = j2 .
i∈I1
2) Denote I2 = {i ∈ Γ(n,q) | (i2 , i2 , i2 + 1) ≤ i ≤
(i2 , q − 1, q − 1) with 0 ≤ i2 ≤ j2 − 1}. For i ∈ I2 ,
we have |Ci | = 3 and |I2 | =
i2 ) =
j23 −3qj22 +(3q2 −1)j2
.
3
(3q 2 − 1)j2 .
j2 −1
i2 =0
Then
i∈I2
(q − i2 − 1)(q −
|Ci | = j23 − 3qj22 +
1073
3) Denote I3 = {i ∈ Γ(n,q) | (j2 , j2 , j2 +1) ≤ i ≤ (j2 , j1 −
1, q − 1)}. For i ∈ I3 , we have |Ci | = 3 and |I3 | =
|Ci | = 3[j22 +
j22 + (1 − q)j2 + (q − 1 − j2 )j1 . Then
i∈I3
(1 − q)j2 + (q − 1 − j2 )j1 ].
4) Denote I4 = {i ∈ Γ(n,q) | (j2 , j1 , j2 + 1) ≤ i ≤
(j2 , j1 , j0 )}. For i ∈ I4 , we have |Ci | = 3 and |I4 | =
|Ci | = 3(j0 − j2 ).
j0 − j2 . Then
i∈I4
Note that I = I1 ∪ I2 ∪ I3 ∪ I4 is a disjoint union. We have
k = n−
i∈I
|Ci | = n −
4
s=1 i∈Is
|Ci | = n − [j23 + (3 − 3q)j22 +
(3q 2 − 3q)j2 + (3q − 3 − 3j2 )j1 + 3j0 ].
IV. C ONCLUDING R EMARKS
In this letter, we investigated the parameters of
HullE (C(q,q3 −1,δ) ), where 2 ≤ δ ≤ n. The reader is
cordially invited to investigate the hulls of primitive narrowsense BCH codes of length q m − 1 with m ≥ 4 and obtain
more self-orthogonal codes.
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